On the Minkowski Unit of Slice Links
Kunio Murasugi · Transactions of the American Mathematical Society · 1965
KUNIO MURASUGI1. Introduction.Let / be an oriented polygonal link of multiplicity u in 3-space R3 and let L be a diagram of /, i.e., an image of a normed regular projection oil onto a plane R2 [9].The orientation of L is inherited from that of/.To L is associated a symmetric integral matrix M, called the symmetric link matrix of / (with respect to L).It is the symmetrized matrix of the Lprincipal minor of the matrix of / [8, §3].If two links are of the same type, then their symmetric link matrices are transformed by a finite sequence of the following two operations and their inverses [8], [10]: Qi : A-''RAR', with R integral and unimodular, 0 Q2 0 0 1 1 01 R' denoting the transposed matrix of R.Now the symmetric link matrix M of a link / may be singular.Then, to M is associated a nonsingular integral matrix B of the highest rank, unless M is a zero matrix (Lemma 2.1).Thus the Minkowski unit CP(B)C) of B can be defined for any prime integer p, including p = (see §2), and is in fact an Received by the editors November 20, 1963.( ) By Cp(B) is meant the Minkowski unit Cp(f) of the quadratic form / associated to B.( ) For definition, see §2.( ) if is a set of all points in R whose *4-coordinate ä 0. 377